scholarly journals The Amazing Chromatic Polynomial

2022 ◽  
Author(s):  
Bruce E. Sagan
Keyword(s):  
Chromatic Polynomial

scholarly journals Algebraic Properties of Chromatic Roots

10.37236/6578 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Peter J. Cameron ◽  
Kerri Morgan
Keyword(s):  
Algebraic Integer ◽  
Tutte Polynomial ◽  
Chromatic Polynomial ◽  
Galois Groups ◽  
Algebraic Numbers ◽  
Isaac Newton ◽  
Isaac Newton Institute ◽  
Chromatic Root ◽  
The Subject ◽  
Chromatic Roots

A chromatic root is a root of the chromatic polynomial of a graph.  Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008.  The purpose of this paper is to report on the seminar and subsequent developments.We conjecture that, for every algebraic integer $\alpha$, there is a natural number n such that $\alpha+n$ is a chromatic root. This is proved for quadratic integers; an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one "interesting" factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.

scholarly journals A Quasisymmetric Function Generalization of the Chromatic Symmetric Function

10.37236/518 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Brandon Humpert
Keyword(s):  
Symmetric Function ◽  
Chromatic Polynomial ◽  
Quasisymmetric Function ◽  
Quasisymmetric Functions ◽  
Acyclic Orientations ◽  
Chromatic Symmetric Function

The chromatic symmetric function $X_G$ of a graph $G$ was introduced by Stanley. In this paper we introduce a quasisymmetric generalization $X^k_G$ called the $k$-chromatic quasisymmetric function of $G$ and show that it is positive in the fundamental basis for the quasisymmetric functions. Following the specialization of $X_G$ to $\chi_G(\lambda)$, the chromatic polynomial, we also define a generalization $\chi^k_G(\lambda)$ and show that evaluations of this polynomial for negative values generalize a theorem of Stanley relating acyclic orientations to the chromatic polynomial.

scholarly journals A result on co-chromatic graphs

1981 ◽  
Vol 4 (2) ◽  
pp. 365-369 ◽  
Author(s):  
E. J. Farrell
Keyword(s):  
Chromatic Polynomial ◽  
Sufficient Condition

A sufficient condition for two graphs with the same number of nodes to have the same chromatic polynomial is given.

Chromatic Solutions

1982 ◽  
Vol 34 (3) ◽  
pp. 741-758 ◽  
Author(s):  
W. T. Tutte
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Chromatic Polynomial ◽  
Basic Theory ◽  
Root Vertex ◽  
Independent Variables ◽  
Image Position ◽  
Formal Power ◽  
Two Independent Variables

Early in the Seventies I sought the number of rooted λ-coloured triangulations of the sphere with 2p faces. In these triangulations double joins, but not loops, were permitted. The investigation soon took the form of a discussion of a certain formal power series l(y, z, λ) in two independent variables y and z.The basic theory of l is set out in [1]. There l is defined as the coefficient of x2 in a more complicated power series g(x, y, z, λ). But the definition is equivalent to the following formula.1Here T denotes a general rooted triangulation. n(T) is the valency of its root-vertex, and 2p(T) is the number of its faces. P(T, λ) is the chromatic polynomial of the graph of T.

scholarly journals An extension of the bivariate chromatic polynomial

2010 ◽  
Vol 31 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Ilia Averbouch ◽  
Benny Godlin ◽  
J.A. Makowsky
Keyword(s):  
Chromatic Polynomial

The Chromatic Polynomial of a Complete r-Partite Graph

1975 ◽  
Vol 82 (7) ◽  
pp. 752 ◽  
Author(s):  
Renu Laskar ◽  
W. R. Hare
Keyword(s):  
Chromatic Polynomial

scholarly journals On the set of Hilbert polynomials

2001 ◽  
Vol 64 (2) ◽  
pp. 291-305 ◽  
Author(s):  
Alexander B. Levin
Keyword(s):  
Chromatic Polynomial ◽  
Hilbert Polynomial ◽  
Graded Algebra ◽  
Open Problems ◽  
Graded Algebras ◽  
Hilbert Polynomials

We characterise the set of all Hilbert polynomials of standard graded algebras over a field and give solutions of some open problems on Hilbert polynomials. In particular, we prove that a chromatic polynomial of a graph is a Hilbert polynomial of some standard graded algebra.

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